$ C = \left[\begin{array}{rr}3 & 1 \\ 3 & -1 \\ -2 & 1\end{array}\right]$ $ v = \left[\begin{array}{r}4 \\ 5\end{array}\right]$ What is $ C v$ ?
Because $ C$ has dimensions $(3\times2)$ and $ v$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ C v = \left[\begin{array}{rr}{3} & {1} \\ {3} & {-1} \\ \color{gray}{-2} & \color{gray}{1}\end{array}\right] \left[\begin{array}{r}{4} \\ {5}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ v$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ v$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ v$ , and so on. Add the products together. $ \left[\begin{array}{r}{3}\cdot{4}+{1}\cdot{5} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ v$ and add the products together. $ \left[\begin{array}{r}{3}\cdot{4}+{1}\cdot{5} \\ {3}\cdot{4}+{-1}\cdot{5} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{3}\cdot{4}+{1}\cdot{5} \\ {3}\cdot{4}+{-1}\cdot{5} \\ \color{gray}{-2}\cdot{4}+\color{gray}{1}\cdot{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}17 \\ 7 \\ -3\end{array}\right] $